Errata

(3) $\Psi$ is defined by the equation $\Psi = Z - \frac{dP}{dt}$, in which (after the explicit differentiation of $P$ with respect to $t$), $x_1$, &c., $y_1$, &c. are to be expressed in terms of the new variables. $y_1$, &c. are thus expressible by the help of the $m$ equations $\frac{dP}{d\xi_i} = \eta_i$ and the $n-m$ equations $\frac{dL}{dt} + \sum_i \left( \frac{dL}{dx_i} \frac{dZ}{dy_i} \right) = 0$. If $(x_1)$, &c., do not contain $t$ explicitly, then $\frac{dP}{dt} = 0$, and $\Psi$ is obtained merely by expressing $Z$ in terms of the new variables. It may be observed that the whole of the above reasoning would apply to the case in which the new variables $\xi_1$, ... $\xi_m$ are more in number than the independent variables of the problem (or $m > n-r$), with this exception; that the $m$ equations $\frac{dP}{d\xi_i} = \eta_i$, together with the $r$ equations obtained by differentiating the equations of condition totally with respect to $t$, would be more than sufficient to express $y_1$, ... $y_n$ in terms of the new variables; consequently $y_1$, &c. might be so expressed in different ways, and therefore, although the value of $\Psi$ obtained by the above rule would certainly be the same as that obtained by recurring to the original formula (D.), the form of $\Psi$ might be different, and therefore the resulting formula erroneous. There must doubtless exist some rule for choosing $n-m$ combinations of the equations of condition in such a way as to lead to the correct forms of $y_1$, ... $y_n$ as functions of the new variables; but I have not at present attempted to investigate it, and perhaps it would be hardly worth while. The theorem in the case in which the new coordinates are independent, may, I believe, be practically useful. ERRATA IN PART I. Art. 1. equation (4.), for $dx$ read $dx_i$. Art. 10. In paragraph preceding equation (26.) omit the words "not containing $t$ explicitly." Art. 18. equation (5), for $y_i$ read $y'_i$. Art. 19. equation (29.), for $h_i$ read $b_i$. Art. 24. second line after equation (L.), for "such as $h$, $k$" read "such as $f$, $g$." Art. 30. The expressions equated to $h$, $k$, $c$, and the three terms in the left-hand column of the table of elements, should each be multiplied by $m$. Art. 42. near the end, for "according as $\Theta$ is between $o$ and $\pi$, or not" read "according as $\Theta$ is between $\pi$ and $2\pi$, or between $o$ and $\pi$."